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Special Functions [1]

Paper Code: 
25DMAT803
Credits: 
6
Contact Hours: 
90.00
Max. Marks: 
100.00
Objective: 

This course will enable the students to

  1. Understand the properties of special functions like Gauss hypergeometric, Legendre functions with their integral representations.
  2. Understand the concept of Bessel’s function, Hermite function with its properties like recurrence relations, orthogonal properties, Generating functions etc.
  3. Understand how special functions are useful in differential equations.

 

Course Outcomes: 

Course

Learning outcomes

(at course level)

Learning and teaching strategies

Assessment

Strategies

Course Code

Course Title

 

 

 

 

25DMAT

803

 

 

 

 

 

Special Functions

(Theory)

 

CO124: Explore the concept of hypergeometric functions, including basic properties, series and integral representations and transformations.

CO125: Classify and explain the functions of different types of differential equations.

CO126: Describe Legender function and   properties such as orthogonality, Recurrence relations, etc.

CO127: Analyse properties such as recurrence relations, Generating functions, Asymptotic expansions and integral representations of Bessel’s function.

CO128: Explore Laguerre and Hermite functions with properties.

CO129: Contribute effectively in course-specific interaction

Approach in teaching:

Discussion, Demonstration, Interactive sessions, Presentation.

 

Learning activities for the students:

Self-learning, Presentation, Effective questions, Assigned tasks

 

Quiz,

Individual and group projects,

Open Book Test, Semester End Examination

 

 

Unit I: 
Gauss hypergeometric functions
12.00
Definition and its properties, Condition of convergence, Integral representation, Gauss theorem, Vandermonde’s theorem, Kummer’s theorem, Linear transformation, Differentiation formulae, Relations of contiguity.
 
Unit II: 
Special caes of Gauss hypergeometric functions
12.00
Gauss’s hypergeometric differential equation and its solution, relation between the solutions of hypergeometric equation, Two summation theorems, Kummer’s confluent hypergeometric function: Definition and differential equation, Integral representation, Differentiation, Kummer’s first and second transformations, contiguous relations.
 
Unit III: 
Legendre polynomials and functions
12.00
Definition, Solution of Legendre’s equation, Legendre functions of the first and second kind, Generating functions(first formula), Rodrigue formula for Pn(x), Orthogonality of Legendre polynomials, Recurrence relations for Pn(x), Beltrami’s result, Christoffel expansion, Christoffel’s summation formula, Relation between Pn(x) and Qn(x), Laplace first and second integrals for Legendre polynomials.
 
Unit IV: 
Bessel functions
12.00
Bessel's equation and its solution, Recurrence relations, Generating function, Integral representations of Bessel’s function, Integrals involving Bessel’s functions.
 
Unit V: 
Hermite polynomials
12.00
Definition, Generating function, Recurrence relations, Orthogonality of Hn(x), Rodrigue formula, Hermite’s differential equation and it’s solution, Laguerre polynomials: Laguerre’s differential equation and it’s solutions, Generating function, Rodrigue formula, Orthogonality of Laguerre polynomials, Recurrence relation.
 
Essential Readings: 
  • R.K. Saxena and D. C. Gokhroo, Special Functions, Jaipur Publishing House, 2014.
  • M. A. Pathan, V. B. L. Chaurasia, J. Banerji and S. P. Goyal, Special Functions and Calculus of Variations, RBD, Jaipur, 2004.
  • E. D. Rainville, Special Functions, Macmillan, New York, 1989.
References: 
  • N. Sneddon, Special Functions, McGraw Hill, New Delhi, 1956.
  • N.N. Lebedev, R. Silverman, Special Functions and Their Application, Dover Publications INC, 2003.
  • Z.X. Wang and D. R. Guo, Special Functions, World Scientific books, 1989.
  • G.E. Andrews, R. Askey and R.  Roy, Special Functions, Cambridge University, 2000.

 

 

e- RESOURCES
 
JOURNALS
 
Academic Year: 
2025-2026 [7]

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Source URL: https://maths.iisuniv.ac.in/node/1455

Links:
[1] https://maths.iisuniv.ac.in/node/1455
[2] https://www.pdfdrive.com/special-functions-their-applications-d164947812.html
[3] https://web.mst.edu/~lmhall/SPFNS/spfns.pdf
[4] https://pkalika.files.wordpress.com/2020/08/special-function-kalika124pages.pdf
[5] https://www.tandfonline.com/toc/gitr20/current
[6] https://www.mdpi.com/journal/mathematics/special_issues/Special_Functions_Applications
[7] https://maths.iisuniv.ac.in/academic-year/2025-2026